172 research outputs found
A Hybrid High-Order method for nonlinear elasticity
In this work we propose and analyze a novel Hybrid High-Order discretization
of a class of (linear and) nonlinear elasticity models in the small deformation
regime which are of common use in solid mechanics. The proposed method is valid
in two and three space dimensions, it supports general meshes including
polyhedral elements and nonmatching interfaces, enables arbitrary approximation
order, and the resolution cost can be reduced by statically condensing a large
subset of the unknowns for linearized versions of the problem. Additionally,
the method satisfies a local principle of virtual work inside each mesh
element, with interface tractions that obey the law of action and reaction. A
complete analysis covering very general stress-strain laws is carried out, and
optimal error estimates are proved. Extensive numerical validation on model
test problems is also provided on two types of nonlinear models.Comment: 29 pages, 7 figures, 4 table
A Hybrid High-Order method for multiple-network poroelasticity
We develop Hybrid High-Order methods for multiple-network poroelasticity, modelling seepage through deformable fissured porous media. The proposed methods are designed to support general polygonal and polyhedral elements. This is a crucial feature in geological modelling, where the need for general elements arises, e.g., due to the presence of fracture and faults, to the onset of degenerate elements to account for compaction or erosion, or when nonconforming mesh adaptation is performed. We use as a starting point a mixed weak formulation where an additional total pressure variable is added, that ensures the fulfilment of a discrete inf-sup condition. A complete theoretical analysis is performed, and the theoretical results are demonstrated on a complete panel of numerical tests
Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods
In this work, we consider the Biot problem with uncertain poroelastic
coefficients. The uncertainty is modelled using a finite set of parameters with
prescribed probability distribution. We present the variational formulation of
the stochastic partial differential system and establish its well-posedness. We
then discuss the approximation of the parameter-dependent problem by
non-intrusive techniques based on Polynomial Chaos decompositions. We
specifically focus on sparse spectral projection methods, which essentially
amount to performing an ensemble of deterministic model simulations to estimate
the expansion coefficients. The deterministic solver is based on a Hybrid
High-Order discretization supporting general polyhedral meshes and arbitrary
approximation orders. We numerically investigate the convergence of the
probability error of the Polynomial Chaos approximation with respect to the
level of the sparse grid. Finally, we assess the propagation of the input
uncertainty onto the solution considering an injection-extraction problem.Comment: 30 pages, 15 Figure
A space-time discontinuous Galerkin method for coupled poroelasticity-elasticity problems
This work is concerned with the analysis of a space-time finite element
discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical
discretization of wave propagation in coupled poroelastic-elastic media. The
mathematical model consists of the low-frequency Biot's equations in the
poroelastic medium and the elastodynamics equation for the elastic one. To
realize the coupling, suitable transmission conditions on the interface between
the two domains are (weakly) embedded in the formulation. The proposed PolydG
discretization in space is then coupled with a dG time integration scheme,
resulting in a full space-time dG discretization. We present the stability
analysis for both the continuous and the semidiscrete formulations, and we
derive error estimates for the semidiscrete formulation in a suitable energy
norm. The method is applied to a wide set of numerical test cases to verify the
theoretical bounds. Examples of physical interest are also presented to
investigate the capability of the proposed method in relevant geophysical
scenarios
A low-order nonconforming method for linear elasticity on general meshes
In this work we construct a low-order nonconforming approximation method for
linear elasticity problems supporting general meshes and valid in two and three
space dimensions. The method is obtained by hacking the Hybrid High-Order
method, that requires the use of polynomials of degree for stability.
Specifically, we show that coercivity can be recovered for by introducing
a novel term that penalises the jumps of the displacement reconstruction across
mesh faces. This term plays a key role in the fulfillment of a discrete Korn
inequality on broken polynomial spaces, for which a novel proof valid for
general polyhedral meshes is provided. Locking-free error estimates are derived
for both the energy- and the -norms of the error, that are shown to
convergence, for smooth solutions, as and , respectively (here,
denotes the meshsize). A thorough numerical validation on a complete panel of
two- and three-dimensional test cases is provided.Comment: 26 pages, 6 tables, and 4 Figure
A serendipity fully discrete div-div complex on polygonal meshes
In this work we address the reduction of face degrees of freedom (DOFs) for
discrete elasticity complexes. Specifically, using serendipity techniques, we
develop a reduced version of a recently introduced two-dimensional complex
arising from traces of the three-dimensional elasticity complex. The keystone
of the reduction process is a new estimate of symmetric tensor-valued
polynomial fields in terms of boundary values, completed with suitable
projections of internal values for higher degrees. We prove an extensive set of
new results for the original complex and show that the reduced complex has the
same homological and analytical properties as the original one. This paper also
contains an appendix with proofs of general Poincar\'e--Korn-type inequalities
for hybrid fields
Numerical modelling of wave propagation phenomena in thermo-poroelastic media via discontinuous Galerkin methods
We present and analyze a high-order discontinuous Galerkin method for the
space discretization of the wave propagation model in thermo-poroelastic media.
The proposed scheme supports general polytopal grids. Stability analysis and
-version error estimates in suitable energy norms are derived for the
semi-discrete problem. The fully-discrete scheme is then obtained based on
employing an implicit Newmark- time integration scheme. A wide set of
numerical simulations is reported, both for the verification of the theoretical
estimates and for examples of physical interest. A comparison with the results
of the poroelastic model is provided too, highlighting the differences between
the predictive capabilities of the two models
A high-order discontinuous Galerkin method for the poro-elasto-acoustic problem on polygonal and polyhedral grids
The aim of this work is to introduce and analyze a finite element
discontinuous Galerkin method on polygonal meshes for the numerical
discretization of acoustic waves propagation through poroelastic materials.
Wave propagation is modeled by the acoustics equations in the acoustic domain
and the low-frequency Biot's equations in the poroelastic one. The coupling is
introduced by considering (physically consistent) interface conditions, imposed
on the interface between the domains, modeling both open and sealed pores.
Existence and uniqueness is proven for the strong formulation based on
employing the semigroup theory. For the space discretization we introduce and
analyze a high-order discontinuous Galerkin method on polygonal and polyhedral
meshes, which is then coupled with Newmark- time integration schemes. A
stability analysis both for the continuous problem and the semi-discrete one is
presented and error estimates for the energy norm are derived for the
semidiscrete problem. A wide set of numerical results obtained on test cases
with manufactured solutions are presented in order to validate the error
analysis. Examples of physical interest are also presented to test the
capability of the proposed methods in practical cases.Comment: The proof of the well-posedness contains an error. This has an impact
on the whole paper. We need time to fix the issu
A Hybrid High-Order method for creeping flows of non-Newtonian fluids
In this paper, we design and analyze a Hybrid High-Order discretization
method for the steady motion of non-Newtonian, incompressible fluids in the
Stokes approximation of small velocities. The proposed method has several
appealing features including the support of general meshes and high-order,
unconditional inf-sup stability, and orders of convergence that match those
obtained for Leray-Lions scalar problems. A complete well-posedness and
convergence analysis of the method is carried out under new, general
assumptions on the strain rate-shear stress law, which encompass several common
examples such as the power-law and Carreau-Yasuda models. Numerical examples
complete the exposition.Comment: 26 pages, 3 figure
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